Global existence and uniqueness of the solution to a nonlinear parabolic equation

Consider the equation $$u^{\mathrm{\prime }}(t)-\mathrm{\Delta }u+\mathrm{\mid }u{\mathrm{\mid }}^{\rho }u=0,u(0)=u_0(x),(1),$$ where $u^{\mathrm{\prime }}:=\frac{du}{dt}$, $\rho =const>0,$ $x\in {\mathbb{R}}^3$, $t>0$. Assume that $u_0$ is a smooth and decaying function, $$\mathrm{\parallel }{u}_0\mathrm{\parallel }=\underset{x\in {\mathbb{R}}^3,t\in {\mathbb{R}}_{+}}{\sup }\mathrm{\mid }u(x,t)\mathrm{\mid }\mathrm{.}$$ It is proved that problem (1) has a unique global solution and this solution satisfies the following estimate $$\mathrm{\parallel }u(x,t)\mathrm{\parallel }<c,$$ where $c>0$ does not depend on $x,t$.